ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

70 4. Two-mode entanglement
0.4
S3 0.2
0
0
0.25
0.5
Μ1
0.75
(a)
1
0.25
0
0.75
0.5
Μ2 Μ2
0.3
0.2
S4
0.1
0
0
0.25
0.5
Μ1
0.75
(b)
1
0.25
0
0.75
0.5
Μ2 Μ2
Figure 4.3. Plot of the nodal surface which solves the equation κp = 0 with
κp defined by Eq. (4.48), for (a) p = 3 and (b) p = 4. The entanglement
of Gaussian states that lie on the leaf–shaped surfaces is fully quantified in
terms of the marginal purities and the global generalized entropy (a) S3 or (b)
S4. The equations of the surfaces in the space Ep ≡ {µ1, µ2, Sp} are given by
Eqs. (4.51–4.53).
Sp for a generic p. Using Maxwell’s relations, we can write
κp ≡ ∂(2˜ν2 −)
∂∆
Sp
= ∂(2˜ν2 −)
∂∆
R
− ∂(2˜ν2 −)
∂R
∆
· ∂Sp/∂∆| R . (4.47)
∂Sp/∂R| ∆
Clearly, for κp > 0 GMEMS and GLEMS retain their usual interpretation, whereas
for κp < 0 they exchange their role. On the node κp = 0 GMEMS and GLEMS
share the same entanglement, i.e. the entanglement of all Gaussian states at κp = 0
is fully determined by the global and marginal p−entropies alone, and does not
depend any more on ∆. Such nodes also exist in the case p ≤ 2 in two limiting
instances: in the special case of GMEMMS (states with maximal global purity at
fixed marginals) and in the limit of zero marginal purities. We will now show that,
besides these two asymptotic behaviors, a nontrivial node appears for all p > 2,
implying that on the two sides of the node GMEMS and GLEMS indeed exhibit
opposite behaviors. Because of Eq. (4.42), κp can be written in the following form
κp = κ2 −
R
˜∆ 2 − R 2
with Np and Dp defined by Eq. (4.44) and
˜∆ = −∆ + 2
µ 2 +
1
2
µ 2 2
,
κ2 =
˜∆
−1 + ,
˜∆ 2 − R2 Np(∆, R)
, (4.48)
Dp(∆, R)
The quantity κp in Eq. (4.48) is a function of p, R, ∆, and of the marginals;
since we are looking for the node (where the entanglement is independent of ∆), we
can investigate the existence of a nontrivial solution to the equation κp = 0 fixing
any value of ∆. Let us choose ∆ = 1 + R 2 /4 that saturates the uncertainty relation